Lisa owns a "Random Candy" vending machine, which is a machine that picks a candy out of an assortment in a random fashion. Lisa controls the probability of picking each candy. The machine has too much of the candy "Coffee Toffee," so Lisa wants to program it so that the probability of getting "Coffee Toffee" twice in a row is greater than $\dfrac{4}{3}$ times the probability of getting a different candy in one try. Write an inequality that models the situation. Use $p$ to represent the probability of getting "Coffee Toffee" in one try.
The strategy We know that Lisa wants to program her "Random Candy" machine so that the probability of getting "Coffee Toffee" twice in a row is greater than $\dfrac{4}{3}$ times the probability of getting a different candy in one try. If we let $C$ denote the probability of getting "Coffee Toffee" twice in a row and $D$ denote the probability of getting a candy other than "Coffee Toffee" in one try, we obtain the inequality $C>\dfrac43D$. Now, let's express $C$ and $D$ in terms of $p$. Expressing the probability of getting "Coffee Toffee" twice in a row We know that the probability of getting "Coffee Toffee" in one try is $p$. Therefore, the probability of getting "Coffee Toffee" twice in a row is $p\cdot p$ or $p^2$. Expressing the probability of getting a candy other than "Coffee Toffee" We know that the probability of getting "Coffee Toffee" in one try is $p$. Therefore, the probability of getting a candy other than "Coffee Toffee" in one try is $(1-p)$. Putting things together We found that $C=p^2$ and $D=1-p$. Since $C>\dfrac43D$, we can substitute and find an inequality in terms of $p$ that models the situation. The answer is: $ p^2>\dfrac43(1-p)$